Abstract
Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r 1,r 2,…,r l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r 1,r 2,⋯ ,r l }-hypergraphs and {r 1,r 2,⋯ ,r l }-hypergraphs concerning the second question.
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Supported in part by National Natural Science Foundation of China (No. 11271116).
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Peng, Y., Wu, B. & Yao, Y. A Note on Generalized Lagrangians of Non-uniform Hypergraphs. Order 34, 9–21 (2017). https://doi.org/10.1007/s11083-016-9385-0
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DOI: https://doi.org/10.1007/s11083-016-9385-0